To ensure that (REF ) has a unique solution, inf-sup stability of \(a_\mu \)  w.r.t. \(V_{H,k,\mu }^{\text{ms}}\)  and \(V_H\) 
is required, which has been shown in [1]}, [2]}.
Compared to these references, we use a slightly different definition of the inf-sup stability constant for (REF )
by using \({}\)  instead of \({}\)  for the test space:
\(\gamma _k:= \inf _{0\ne w_H\in V_H}\sup _{0\ne v_H\in V_H}\frac{a_\mu (w_H-(w_H), v_H)}{{w_H-(w_H)}{v_H}}.\) 
